(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. This implies Q-termination of R.
The following rules were used to construct the certificate:
f(active(X)) → f(h(g(mark(X))))
f(active(X)) → active(f(X))
h(active(X)) → active(h(X))
mark(f(X)) → f(mark(X))
mark(h(X)) → h(mark(X))
f(proper(X)) → proper(f(X))
g(proper(X)) → proper(g(X))
h(proper(X)) → proper(h(X))
ok(f(X)) → f(ok(X))
ok(g(X)) → g(ok(X))
ok(h(X)) → h(ok(X))
mark(top(X)) → proper(top(X))
ok(top(X)) → active(top(X))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 34, 35, 36, 37, 38, 40, 41, 44, 45, 48, 49, 50, 51, 58, 60, 61, 63, 64, 65
Node 1 is start node and node 3 is final node.
Those nodes are connected through the following edges:
- 1 to 5 labelled f_1(0), g_1(0), h_1(0)
- 1 to 8 labelled active_1(0), proper_1(0)
- 1 to 7 labelled f_1(0), h_1(0)
- 1 to 20 labelled f_1(1)
- 1 to 23 labelled active_1(1), proper_1(1)
- 1 to 58 labelled proper_1(2)
- 3 to 3 labelled #_1(0)
- 5 to 6 labelled h_1(0)
- 5 to 3 labelled ok_1(0)
- 5 to 13 labelled f_1(1), g_1(1), h_1(1)
- 5 to 14 labelled active_1(1)
- 5 to 34 labelled proper_1(1)
- 5 to 35 labelled f_1(2)
- 5 to 38 labelled active_1(2)
- 5 to 63 labelled proper_1(3)
- 5 to 64 labelled proper_1(2)
- 6 to 7 labelled g_1(0)
- 6 to 25 labelled proper_1(1)
- 7 to 3 labelled mark_1(0)
- 7 to 15 labelled f_1(1), h_1(1)
- 7 to 14 labelled proper_1(1)
- 7 to 38 labelled proper_1(2)
- 8 to 3 labelled f_1(0), h_1(0), g_1(0), top_1(0)
- 8 to 16 labelled f_1(1)
- 8 to 19 labelled active_1(1), proper_1(1)
- 8 to 48 labelled proper_1(2)
- 13 to 3 labelled ok_1(1)
- 13 to 13 labelled f_1(1), g_1(1), h_1(1)
- 13 to 14 labelled active_1(1)
- 13 to 35 labelled f_1(2)
- 13 to 38 labelled active_1(2)
- 13 to 63 labelled proper_1(3)
- 13 to 64 labelled proper_1(2)
- 14 to 3 labelled top_1(1)
- 15 to 3 labelled mark_1(1)
- 15 to 15 labelled f_1(1), h_1(1)
- 15 to 14 labelled proper_1(1)
- 15 to 38 labelled proper_1(2)
- 16 to 17 labelled h_1(1)
- 16 to 44 labelled proper_1(2)
- 17 to 18 labelled g_1(1)
- 17 to 40 labelled proper_1(2)
- 18 to 3 labelled mark_1(1)
- 18 to 15 labelled f_1(1), h_1(1)
- 18 to 14 labelled proper_1(1)
- 18 to 38 labelled proper_1(2)
- 19 to 3 labelled f_1(1), h_1(1), g_1(1)
- 19 to 16 labelled f_1(1)
- 19 to 19 labelled active_1(1), proper_1(1)
- 19 to 48 labelled proper_1(2)
- 20 to 21 labelled h_1(1)
- 20 to 49 labelled proper_1(2)
- 21 to 22 labelled g_1(1)
- 21 to 45 labelled proper_1(2)
- 22 to 14 labelled mark_1(1)
- 22 to 41 labelled proper_1(2)
- 22 to 38 labelled mark_1(1)
- 22 to 37 labelled f_1(2), h_1(2)
- 22 to 60 labelled proper_1(3)
- 23 to 14 labelled f_1(1), h_1(1)
- 23 to 34 labelled f_1(1), g_1(1), h_1(1)
- 23 to 38 labelled f_1(1), h_1(1)
- 23 to 63 labelled f_1(1), g_1(1), h_1(1)
- 23 to 64 labelled f_1(1), g_1(1), h_1(1)
- 25 to 14 labelled g_1(1)
- 25 to 38 labelled g_1(1)
- 34 to 25 labelled h_1(1)
- 35 to 36 labelled h_1(2)
- 35 to 61 labelled proper_1(3)
- 36 to 37 labelled g_1(2)
- 36 to 51 labelled proper_1(3)
- 37 to 14 labelled mark_1(2)
- 37 to 38 labelled mark_1(2)
- 37 to 41 labelled proper_1(2)
- 37 to 50 labelled f_1(3), h_1(3)
- 37 to 60 labelled proper_1(3)
- 37 to 65 labelled proper_1(4)
- 38 to 14 labelled f_1(2), h_1(2)
- 38 to 38 labelled f_1(2), h_1(2)
- 40 to 14 labelled g_1(2)
- 40 to 38 labelled g_1(2)
- 41 to 3 labelled top_1(2)
- 44 to 40 labelled h_1(2)
- 45 to 41 labelled g_1(2)
- 45 to 60 labelled g_1(2)
- 48 to 44 labelled f_1(2)
- 49 to 45 labelled h_1(2)
- 50 to 14 labelled mark_1(3)
- 50 to 38 labelled mark_1(3)
- 50 to 41 labelled proper_1(2)
- 50 to 50 labelled f_1(3), h_1(3)
- 50 to 60 labelled proper_1(3)
- 50 to 65 labelled proper_1(4)
- 51 to 41 labelled g_1(3)
- 51 to 60 labelled g_1(3)
- 51 to 65 labelled g_1(3)
- 58 to 49 labelled f_1(2)
- 60 to 41 labelled f_1(3), h_1(3)
- 60 to 60 labelled f_1(3), h_1(3)
- 60 to 65 labelled f_1(3), h_1(3)
- 61 to 51 labelled h_1(3)
- 63 to 61 labelled f_1(3)
- 64 to 63 labelled f_1(2), g_1(2), h_1(2)
- 64 to 64 labelled f_1(2), g_1(2), h_1(2)
- 65 to 60 labelled f_1(4), h_1(4)
- 65 to 65 labelled f_1(4), h_1(4)